Approximation Algorithms for Orienteering and Discounted-Reward TSP

In this paper, we give the first constant-factor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the Discounted-Reward TSP, motivated by robot navigation. In both problems, we are given a graph with lengths on edges and prizes (rewards) on nodes, and a start node s. In the Orienteering Problem, the goal is to find a path that maximizes the reward collected, subject to a hard limit on the total length of the path. In the Discounted-Reward TSP, instead of a length limit we are given a discount factor γ, and the goal is to maximize total discounted reward collected, where reward for a node reached at time t is discounted by γ . This problem is motivated by an approximation to a planning problem in the Markov decision process (MDP) framework under the commonly employed infinite horizon discounted reward optimality criterion. The approximation arises from a need to deal with exponentially large state spaces that emerge when trying to model one-time events and non-repeatable rewards (such as for package deliveries). We also consider tree and multiplepath variants of these problems and provide approximations for those as well. Although the unrooted orienteering problem, where there is no fixed start node s, has been known to be approximable using algorithms for related problems such as k-TSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem [3, 1]. We complement our approximation result for Orienteering by showing that the problem is APX-hard. Computer Science Department, Carnegie Mellon University. Research supported by NSF Grants CCR-0105488, IIS-0121678, and CCR-0122581. email: {avrim,shuchi}@cs.cmu.edu MIT Computer Science and Artificial Intelligence Laboratory (CSAIL). email: [email protected] Department of Computer Science, University of New Mexico. email: [email protected] Computer Science Department, University of California, Los Angeles. This work was done while the author was visiting Carnegie Mellon University. email: [email protected] Max-Planck-Institut fr Informatik. This work was done while the author was a graduate student at MIT. email: [email protected]